Algebraic Manifolds Research Articles

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

Algebra environments II. Algebra homomorphisms and derivations

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms.

Symplectic complexity of reductive group actions

Let a complex algebraic reductive group G act on a complex algebraic manifold X. For a G-invariant subvariety Ξ of the nilpotent cone N(g∗)⊂g∗ we define a notion of Ξ-symplectic complexity of X. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of Ξ-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.

Parametric Expansions of an Algebraic Variety Near Its Singularities II

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity.

Computing Geometric Feature Sizes for Algebraic Manifolds

Computing Geometric Feature Sizes for Algebraic Manifolds

Invariant Algebraic Manifolds for the Rucklidge Model of Double Convection

Invariant Algebraic Manifolds for the Rucklidge Model of Double Convection

Localization measures of parity adapted U(D)-spin coherent states applied to the phase space analysis of the D-level Lipkin-Meshkov-Glick model.

We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N→∞ limit. The D=3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D=2 atomic coherent states, to define the coherent state representation Q_{ψ} (Husimi function) of a symmetric N-qudit state |ψ〉 in the phase space CP^{D-1} (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N→∞ limit, where the discrete parity symmetry Z_{2}^{D-1} is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2^{D-1} different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.

On singular generalizations of the Singer–Hopf conjecture

AbstractThe Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.

The probabilistic vs the quantization approach to Kähler–Einstein geometry

In the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic manifold X—involving random point processes on X—a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler–Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler–Einstein geometry.

Foliated affine and projective structures

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

Simply connected Voisin manifolds of dimension four

AbstractVoisin constructed a series of examples of simply connected compact Kähler manifolds of even dimension, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we prove that Voisin's examples of dimension four also do not have the rational homotopy type of a complex projective manifold. Oguiso constructed simply connected compact Kähler manifolds starting from dimension four, which cannot deform to a complex projective manifold under a small deformation. We also prove that Oguiso's examples do not have the rational homotopy type of a complex projective manifold.

Chow transformation of coherent sheaves

Abstract We define a dual of the Chow transformation of currents on any complex projective manifold. This integral transformation is a factor of a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear differential operator, which does not commute with ∂ \partial or ∂ ¯ \overline{\partial } . We obtain a complete intrinsic resolution of the problem of the algebraicity of the cohomology classes. On another hand, in the case of the complex projective space, we give the translation in terms of real-analytic D {\mathcal{D}} -modules of the properties of the Chow transformation. Then, the proofs can be simplified by using the conormal currents, which exist for all currents of bidimension ( p , p ) \left(p,p) on the complex projective space, even not closed. This is a consequence of the existence of dual currents, defined on the dual complex projective space. In particular, we obtain a linear differential system of order lower than that of the Gelfand-Gindikin-Graev differential system, characterizing the images by the Chow transformation of smooth differential forms on the complex projective space.

Quantum Correlations in the Minimal Scenario

In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted Q, is fundamental for quantum information theory. We review and systematize what is known about Q, and add many details, visualizations, and complete proofs. In particular, we provide a detailed description of the boundary, which consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, i.e., the application of the sine function to each coordinate. This transforms the classical correlation polytope exactly into the correlation body Q, also identifying the boundary structures. The second principle, self-duality, is an isomorphism between Q and its polar dual, i.e., the set of affine inequalities satisfied by all quantum correlations (“Tsirelson inequalities''). The same isomorphism links the polytope of classical correlations contained in Q to the polytope of no-signalling correlations, which contains Q.We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables, and establish a new non-linear inequality for Q involving the determinant of the correlation matrix.

Tri-Sectional Approximation of the Shortest Path to Long-Term Voltage Stability Boundary With Distributed Energy Resources

Ensuring long-term voltage stability is critical for reliable operations of power grids. High share of distributed energy resources (DERs) can create complicated system operation modes that may invalidate the traditional long-term voltage stability analysis based on typical operation modes. To address this challenge, this paper investigates how to compute the shortest path to the voltage stability boundary in the DER aggregated load space with large dispersion. Instead of working in the Euclidean space, we establish the analysis and computations on the algebraic power flow manifold to better capture the curvature change of the shortest path along the direction of losing stability. A modified optimal control framework is presented for obtaining the ground-truth of the smooth shortest path on the manifold. To efficiently and accurately solve for the shortest path, we further leverage the geometric features of the power flow manifold and propose a tri-sectional approximation model that is scalable for large-scale systems. Several numerical examples, up to the 1354-bus system, with different DER penetration levels and high dimensional renewable power injection variations are evaluated. The simulation results demonstrate that the tri-sectional approximation achieves high accuracy and efficiency to approximate the shortest path to the voltage stability boundary.

К ПРОЕКТИВНО–ДИФФЕРЕНЦИАЛЬНОЙ ГЕОМЕТРИИ ПЯТИМЕРНЫХ КОМПЛЕКСОВ ДВУМЕРНЫХ ПЛОСКОСТЕЙ ПРОЕКТИВНОГО ПРОСТРАНСТВА P5

The article focuses on the differential geometry  of 5-dimensional complexes  C5 of 2-dimensional planes in the projective  space  P5 that contains a finite  number of developable surfaces. This article relates to researches on the projective differential geometry based on E. Cartan’s moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint and also extend the results to wider classes of manifolds of multidimensional planes. In order to study such submanifolds, we apply the Grassmann mapping of the manifold  G(2, 5) onto the 9-dimensional algebraic manifold Ω(2, 5) of the space P19. The main task of the differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, determine their structure and define the degree of the arbitrariness of their existence and also to research properties of submanifolds of various classes. Intersection of the algebraic manifold Ω(2, 5) with its tangent space TlΩ(2, 5) represents the Segre cone Cl(3, 3). This 5-dimensional cone carries two sets of plane 3-dimensional generatrices intersecting in straight lines. The projectivization PBl(2) of this cone is the Segre manifold Sl(2, 2). The Segre manifold Sl(2, 2) is invariant under projective transformations of P8 = PTlΩ(2, 5), which is the projectivization with center at the point l of the tangent space TlΩ(2, 5) to the algebraic manifold Ω(2, 5). The Segre manifold Sl(2, 2) is used for classification of the submanifolds of Grassmann manifold G(2, 5) under consideration, as well as for interpretation of their properties  in the projective algebraic manifold terms.  The classification  of submanifolds of the Grassmann manifold G(2, 5) is  based on  various  configurations of the plane PTlΩ(2, 5) and on the Segre manifold Sl(2, 2).  The goal of this article  is to prove geometrically the theorem for determination of the order of the Segre invariant manifold Sl(2, 2).

A RIGIDITY PROPERTY OF PLURIHARMONIC MAPS FROM PROJECTIVE MANIFOLDS

AbstractSuppose M is a complex projective manifold of dimension $\geq 2$ , V is the support of an ample divisor in M and U is an open set in M that intersects each irreducible component of V. We show that a pluriharmonic map $f:M\to N$ into a Kähler manifold N is holomorphic whenever $f\vert _{V\,\cap \, U}$ is holomorphic.

Matrix compression along isogenic blocks

A matrix-compression algorithm is derived from a novel isogenicblock decomposition for square matrices. The resulting compression andinflation operations possess strong functorial and spectral-permanenceproperties. The basic observation that Hadamard entrywise functionalcalculus preserves isogenic blocks has already proved to be of paramountimportance for thresholding large correlation matrices. The proposedisogenic stratification of the set of complex matrices bears similarities tothe Schubert cell stratification of a homogeneous algebraic manifold. Anarray of potential applications to current investigations in computationalmatrix analysis is briefly mentioned, touching concepts such as symmetricstatistical models, hierarchical matrices and coherent matrix organizationinduced by partition trees.

Lagrangian Geometry of Algebraic Manifolds

Lagrangian Geometry of Algebraic Manifolds

Degenerating Kähler–Einstein cones, locally symmetric cusps, and the Tian–Yau metric

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of K\"ahler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $\omega_\beta$ converges to the locally symmetric metric and further give asymptotics of $\omega_\beta$ when $X\setminus D$ is a ball quotient. Our second result deals with the case when $X$ is Fano and $D$ is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of $\omega_\beta$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,\omega_\beta)$ converges to an interval in the Gromov-Hausdorff sense.